Numerical investigations for flow past two square rods in staggered arrangement through Lattice Boltzmann method

Cd: Drag force; Cl: Lift force; Cdmean: Mean drag coeffi cient; Cdrms: Root-mean-square value of drag coeffi cient; Clrms: Rootmean-square value of lift coeffi cient; FEM: Finite Element Method; CFD: Computational Fluid Dynamics; D: Size of main rod; d: Size of control rod; Ωd: In-line force component; ei: Velocity vectors; ƕi: Particle distribution function; ƕi(eq): Equilibrium distribution function; Ϗs: Vortex shedding frequency; g: Gap spacing; H: Height of channel; L: Length of channel; Ld: Downstream position; Lu: Upstream position; ᴨ: Number of particles; St: Strouhal number; U∞:Uniform infl ow velocity; ξi: Weighting coeffi cients; FFT: Fast Fourier Transform Greek symbols τ: Relaxation-time; γ: Kinematic viscosity; ρ: Fluid density

but also in civil and industrial scale such as offshore, pillars, cables, sky scrapers, vertical columns of a platform in the sea, towers and river crossing bridges. Structural design fl ow induced vibrations such as cooling of electronic equipment and acoustic emissions are some other examples of engineering fi elds. The separation of periodic vortex shedding and fl uid fl ow from the structures produces drag and lift forces which cause structure damages and great loss of energy. Therefore it is important to control the harmful effect of fl uid fl ow. For this reason many researchers have paid much attention to suppress vortex shedding and weakening the lift and drag force. The wake and the separated fl ow region can be reduced by fl ow control methods such as active and passive control methods which, in turn, reduces the drag force and suppress the vortex shedding. Active control method uses external energy to control the fl ow such as jet blowing and forced fl uctuations [1] and Fujisawa [2]. Passive control method either modify the shape of the body or attach or detach a control rod to the main body in order to control the fl ow. Inhomogeneous inlet fl ow, control rod, end plate, a vertical plate placed upstream of the body in the shear layer are examples of passive control method [3]. A variety of ways have been used to suppress the fl uid forces acting on the bluff bodies in passive control methods. Various experimental and numerical studies are accessible in literature based on circular, square and rectangular rods in passive control methods for purpose of reducing fl uid forces and the suppression of vortex shedding. Williamson [4] determined the fl ow around two square rods by taking Re = 50 -200 and gap spacing between the rods within the range of g = 0.5 -5 by applying Visualization Method. He analyzed that fl ow regime between the rods are in phase and anti-phase due to the effect of the gap spacing. Kawanura, et al. [5] performed numerical simulations for fl ow past over a single square rod through applying Finite Difference Method (FDM) at Reynolds number, Re = Re= 1 0 3 -10 5 . It was found that Cd mean of rod with rough surfaces reduces sharply at Re= 2×10 4 , that is known as critical Reynolds number. Zdravkovich [6] conducted a numerical study by considering the arrangement of rods in three distinct ways; inline setting, staggered setting and side by side setting. He investigated that the gap spacing between the rods plays essential role in determining the fl ow regimes. Mansingh and Oosthuizen [7] experimentally studied the control rod effect placed downstream of a rectangular rod for different plate lengths over a range of Reynolds numbers from 350 to 1150. They found that the Strouhal number decreases in presence of downstream control rod. Park and Higuchi [8] discussed the fl ow regime in a wake region by using the short rod placed near to the main rod. An instantaneous fl ow regime was observed close to main rod in the wake area. It was observed that vortex shedding and drag force from the base of rod also decreased, that was due to the suppression of vortex shedding in near wake region. Alam, et al. [9] studied how to reduce the fl uid forces acting upon a square triangular rod that are arranged in tandem arrangement. They reported that fl uid forces are controlled by changing the distance between the control rod and upstate square triangular rod. Texier, et al. [10] studied the behavior of fl ow over a semicircular rod in the presence of control rod at Re = 200 -400. The secondary rod vortex shedding are introduced in the fl ow fi eld by taking zero velocity called stagnation zone. Dutta, et al. [11] considered low Reynolds number to study fl ow behavior for fl ow past over a square rod. They showed that vorticity decreased faster at low Reynolds number in downstate direction as compared to the high Reynolds number. Zhou, et al. [12] numerically studied the fl uid force reduction acting on square rod in two dimensional fl ow by utilizing control rod that is placed at upstream direction through applying Lattice Boltzmann Method (LBM). The investigation was perfumed to examine the abatement in liquid powers. It was revealed that drag coeffi cient (Cd rms ) decreases and control of vortex shedding happened in stream fi eld. Agrawal, et al. [13] analyzed the effect of gap spacing (g = 0.7 -2.5) for a fl ow past over two square rods in staggered arrangement by applying LBM at Re = 73. They observed fl ip fl opping and synchronized fl oe regimes. Furthermore, they showed that the strength of above mention fl ow regime are largely depends upon the gap spacing. Shao and Wai [14] investigated the reduction of vortex shedding for fl ow around a square object for the limitation of Reynolds number (Re) up-to two digits. Furthermore, they enhanced the arrangements and used small square rod. It was shown that complete vortex shedding depends upon bluffness of an object which is a square rod. Guo, et al. [15] considered two dimensional stream fl ow over a square object with Reynolds number at Re = 10-300 by applying Finite Difference Method (FDM). The required results are yields from Lattice Boltzmann equation and gas kinetic technique. It was reported that the grid Boltzmann condition is quicker than gas active plan for consistent and shaky stream, as gas kinetic technique gives better yield to certain cases, which Lattice Boltzmann equation didn't give. Turki, (2008) [16] determined the impacts of control rod set both joined and disconnected arrangement around a square chamber by applying Control Volume Finite Element Method (CVFEM). The Reynolds number are taken within the range of 40 < Re < 200, in which the drag coeffi cients (Cd rms ), lift coeffi cients Clrms and Strouhal number S t are calculated. Vikram, et al. [17] experimentally observed two dimensional non steady fl ow past over two square rods with an inline setting in a free stream. They showed that vortex shedding frequency is same between the rods as well in downstream of the rod. It was also observed that upstream rod has higher lift coeffi cient as compared to downstream rod. Yen and Liu [18] analyzed the fl ow behavior of a square rod and two control rods which are arranged in staggered arrangement at open loop wind tunnel. The Reynolds number are taken as 2262≤Re≤2800 and gap spacing ratio is selected as 0 ≤ g * ≤ 12. Here three different fl ow regimes are observed; single regime, gap fl ow regime and couple vortex shedding. Largest value of drag coeffi cient is observed at single regime and smallest value of drag coeffi cient is obtained at gap fl ow regime. It was also observed that largest value of S t is obatined at single fl ow regime. Ali, et al. [19] numerically explored the stream structure past over a rectangular chamber isolates with two control rods of different lengths at Re = 150.
The control rods length was (L). The stream conduct can be grouped in three unique systems. In fi rst system the length of the rod is short (0 ≤ L ≤ l), at that point the free shear layers is sentenced. In second system the length of the rod is medium Method is given by [29]; where ƕ t is the particle distribution function at position x and time t, ƕ eq t is the corresponding equilibrium distribution function of i th discrete particle velocity u, i is the direction of velocity and τ is the relaxation time.
The equilibrium distribution function is computed as below [30].
Here ) is the speed of sound [31]. The fl ow velocity u and density ρ can be obtained by ρ = Σ ƕ t and ρu = Σ ƕ t e i i = 0, 1, 2, 3,…, 8 (5) LBM based on different lattice models, depending on the problem, for fl uid fl ow simulation (Guo, (2008)). But the most commonly used model for two-dimensional (2-D) fl ow problems is the two-dimensional nine-velocityparticles model (D2Q9). In present study, we also used D2Q9 model. Where, eight particles are moving along their axis and diagonally and one is rest particle (Figure 1).

Statement of problem and boundary conditions
The schematic confi guration of fl ow past over two square rods of same sizes D = 20 in staggered arrangement detached with rectangular control rod of size d= 10 is shown in Figure 2 The Since the rods are stationary so, no-slip boundary conditions are used on top and bottom walls of channels as well on the surfaces of rods [15]. The uniform infl ow velocity is used at entrance of domain and convective boundary condition is applied at the exit position of channel [32]. The total fl uid forces on rectangular rods are calculated using the momentum exchange method [32].

Important parameters
Some important physical parameters are used in this paper which are defi ned as following   Table 1.

Effect of computational domain, uniform infl ow velocity and code validation study
To study the effect of selected computational domain on numerical simulations, different grid resolutions are employed ( Table 2) [34]. Similarly, the values of Cdmean obtained at Re = 200 from present result is most approximately similar value of Cd mean obtained from the results of Okajima [35], Gera, et al. [34] and Norberg [36]. In this way the values of Strouhal number at Re = 100, 150 and 200, obtained present results having close agreement with results obtained from data of Norberg [36], Davis and Moore [37] and Robichuax, et al. [38].

Results and Discussions
In this section, the effect of gap spacing (g = 1, 3 & 6) between the rods and Reynolds number (Re = 80 -200) on fl ow structure mechanism is discussed for fl ow past over two square rods in staggered confi guration. The results are obtained in terms of vorticity visualization, power spectrum analysis of lift coeffi cients and force statistics. It is important to state here that in vorticity contour visualization graphs the solid lines represent the positive vortices generated from the lower corner and dashed lines represent the negative vortices generated from the upper corner of the rods. To avoid repetition only some important representative plots will be shown in this paper. Since the rapid buildup of vortex shedding defi ne the phenomena of oscillation. The oscillating fl ow mechanism is described by Strouhal number which is calculated by applying the Fast Fourier Transform (FFT) on lift coeffi cients data. The highest peak in each spectrum graph is termed as primary vortex shedding frequency (PVSF). These PVSF give the value of Strouhal number for each case. The other peaks in the spectrum graphs are termed as secondary rod interaction

Irregular Single Bluff Body Flow Regime (ISBB)
The fi rst fl ow regime is observed in current study is   is also notifi ed that power spectrum at g = 3, Re = 100, 125, 175, 200 provided the single sharp peak called primary peak. The rod R 2 containing maximum magnitude than the Rod R 1 . The magnitude of power spectrum analysis is also increases with increase in Re ( Figure 6(g, h)).

Anti-Phase Synchronized Flow Regime
The Anti-Phase Synchronized fl ow regime is observed at g = 6 with Re = 80-200. The interaction between the wakes is   Table 4, Figure 8(a-h).

Force statistics
The variation of aerodynamic force coeffi cients with gap spacing, g = 1, 3 & 6 at Re = 80-200 is presented in Figure 6    is Irregular Single Bluff Body (ISBB) fl ow regime. Now if we increase the gap spacing from g = 1 to 3 and 6, the value of S t for both the rods, is representing the mixed trend from Re = 80 -200. The maximum value of S t is obtained for second rod R 2 with g = 6 and Re = 150, that is 0.1742 as shown in Figure 6(d), where existing fl ow regime is Anti-Phase Synchronized (APS) fl ow regime. All calculated values of physical parameters, are also shown in Figure 9 Table 5.

Conclusions
The present numerical study represented the analysis The mean drag coeffi cient of rod R 1 is greater than the Cd mean of rod R 2 for all three selected gap spacing. It is also representing increasing and decreasing behavior with increment in value of Reynolds number.
The maximum value of mean drag coeffi cient is attained for   The values of Strouhal number at g = 1 for both the rods are less than the values of S t at g = 3 and 6 for rod R 1 and R 2 .